Lecture Notes in Computer Science 3472

1 Homing and Synchronizing Sequences 31
Length of Synchronizing Sequences and Čern´y’s Conjecture. Synchronizing
sequences were introduced a bit later by Čern´y[Čer64] and studied mostly
independently from homing sequences, with some exceptions [Koh78, Rys83,
LY96]. The focus has largely been on the worst-case length of sequences, except
an article by Rystsov that classifies the complexity of several related problems
[Rys83], the article by Eppstein, which introduces the algorithm in Section 1.3.2
[Epp90], and the survey by Lee and Yannakakis [LY96]. Čern´y [Čer64] showed
an upper bound of 2n − n − 1 for the length of synchronizing sequences and
conjectured that it can be improved to (n − 1) 2 , a conjecture that inspired much
of the research in the area. The first polynomial bound was 1
2n3 − 3
2n2 + n +1
due to Starke [Sta66], and as mentioned in Section 1.3.2 the best known bound is
1
6 (n3 −n) due to Klyachko, Rystsov and Spivak [KRS87]. Already Čern´y [Čer64]
proved that there are automata that require synchronizing sequences of length
at least (n − 1) 2 , so if the conjecture is true then it is optimal.
Proving or disproving Čern´y’s conjecture is still an open problem, but it has
been settled for several special cases: Eppstein [Epp90] proved it for monotonic
automata, which arise in the orientation of parts that we saw in Example 1.3;
Kari [Kar03] showed it for Eulerian machines (i.e., where each state has the
same in- and out-degrees); Pin [Pin78b] showed it when n is prime and the
machine is cyclic (meaning that there is an input letter a ∈ I such that the
a-transitions form a cycle through all states); Čern´y, Pirická and Rosenauerová
[ ČPR71] showed it when there are at most 5 states. Other classes of machines
were studied by Pin [Pin78a], Imreh and Steinby [IS95], Rystsov [Rys97], Bogdanović
et al. [BIĆP99], Trakhtman [Tra02], Göhring [Göh98] and others. See
also Trakhtman’s [Tra02] and Göhring’s [Göh98] articles for more references.
Parallel Algorithms. In a series of articles, Ravikumar and Xiong study the
problem of computing homing sequences on parallel computers. Ravikumar gives
a deterministic O( √ n log 2 n) time algorithm [Rav96], but it is reported not to
be practical due to large communication costs. There is also a randomized algorithm
requiring only O(log 2 n) time but O(n 7 ) processors [RX96]. Although not
practical, this is important as it implies that the problem belongs to the complexity
class RNC. The same authors also introduced and implemented a practical
randomized parallel algorithm requiring time essentially O(n 3 /k), where the
number k of processors can be specified [RX97]. It is an open problem whether
there are parallel algorithms for the synchronizing sequence problem, but in the
special case of monotonic automata, Eppstein [Epp90] gives a randomized parallel
algorithm. See also Ravikumar’s survey of parallel algorithms for automata
problems [Rav98].
Nondeterministic and Probabilistic Automata. The homing and synchronizing
sequence problems become much harder for some generalizations of Mealy
machines. As shown in Exercise 1.8, they are PSPACE-complete for nondeterministic
automata, where δ(s, a) is a subset of S. This was noted by Rystsov
[Rys92] as a consequence of the PSPACE-completeness theorem in another of